Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Spectral Theory on Hyperbolic Surfaces
David Borthwick
Emory UniversityJuly, 2010
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Outline
Hyperbolic geometry
Fuchsian groups
Spectral theory
Selberg trace formula
Arithmetic surfaces
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
I. Hyperbolic Geometry
(Escher/Jos Leys)
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Mobius transformations
The upper half-plane H = {Im z > 0} has a large group of conformal
automorphisms, consisting of Mobius transformations of the form
z 7→ az + b
cz + d,
where a, b, c, d ∈ R and ad− bc > 0.
These symmetries form the group
PSL(2, R) := SL(2, R)/{±I}
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Hyperbolic metric
Under the PSL(2, R) action, H has an invariant metric,
ds2 =dx2 + dy2
y2,
often called the Poincare metric.
This metric is hyperbolic, meaning that the Gauss curvature is −1.
We have a corresponding measure
dA(z) =dx dy
y2,
and distance function
d(z, w) = log|z− w|+ |z− w||z− w| − |z− w|
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Geometry in H
The hyperbolic metric is conformal to the Euclidean metric, so angles
are computed as in Euclidean geometry.
Geodesics are arcs of generalized circles intersecting ∂H := R ∪ {∞}orthogonally.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
The Gauss-Bonnet theorem gives a formula for areas of triangles with
geodesic sides.
α
α
ββ
γ
γ
Area(ABC) = π − α− β − γ.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Fixed points
Elements of PSL(2, R) are classified according to their fixed points.
For g ∈ PSL(2, R), the fixed point equation z = gz is quadratic:
cz2 + (d − a)z− b = 0
For each g there are exactly 2 solutions in C ∪∞.
1. elliptic: one fixed point in H. (The other must be the complex
conjugate.) An elliptic transformation is a rotation centered at the
fixed point.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
2. parabolic: a single degenerate fixed point (must lie on ∂H.) Anyparabolic transformation is conjugate to the map z 7→ z + 1.
3. hyperbolic: two distinct fixed points in ∂H. A hyperbolic
transformation is conjugate to the map z 7→ eℓz for some ℓ ∈ R.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
II. Fuchsian groups
A Fuchsian group is a discrete subgroup Γ ⊂ PSL(2, R) (“discrete” in
the matrix topology, equivalent to Euclidean R4.)
This is equivalent to the condition that Γ acts properly discontinuously
on H, meaning that each orbit Γz are locally finite.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Quotients
The quotient Γ\H is a surface whose points correspond to the orbits of
Γ in H. For Fuchsian Γ, the quotient inherits a hyperbolic metric fromH.
In general, Γ\H is an orbifold. The quotient is a smooth surface iff Γhas no elliptic elements.
A fundamental domain for Γ is a closed region F such that the
translates of F under Γ tesselate H.
F
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Automorphic forms
A function f on Γ\H is equivalent to a function on H satisfying
f(gz) = f(z) for g ∈ Γ.
The latter is called an automorphic function for Γ.
An automorphic form of weight k satisfies
f(gz) = (cz + d)kf(z).
(These are sections of the k-th power of the canonical line bundle over
Γ\H.)
Warning: in some contexts automorphic forms are required to be
meromorphic or even entire.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Uniformization
[Koebe & Poincare, 1907]: For any smooth complete Riemannianmetric on a surface, there is a conformally related metric of constant
curvature. (By scaling, we can restrict this curvature to the values
−1, 0, 1.)
Up to isometry, the only simply connected possibilities are the sphere
S2, Euclidean R2, and the hyperbolic plane H
2.
All of the surfaces with Euler characteristic χ < 0 (i.e. most surfaces)are uniformized by hyperbolic quotients Γ\H.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Riemann surfaces
The term hyperbolic refers to curvature = −1. But because thehyperbolic isometries of H are the same as the conformal
automorphisms, any quotient Γ\H has a natural complex structure.
A Riemann surface is a one-dimensional complex manifold.
Uniformization implies that any Riemann surface can be realized as a
quotient of the Riemann sphere C ∪ {∞}, the complex plane C, and
the upper half-plane H.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Limit set
The limit set Λ(Γ) of Γ ⊂ PSL(2, R) is the collection of limit points oforbits of Γ, in the Riemann sphere topology.
In fact, it suffices to take the orbit of any single point that’s not an
elliptic fixed point.
Λ(Γ) is a closed, Γ-invariant subset of ∂H = R ∪ {∞}.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Classification of Fuchsian groups
[Poincare, Fricke-Klein]: Fuchsian groups are classified as
1. Elementary: Λ(Γ) contains 0, 1, or 2 points;
2. First Kind: Λ(Γ) = ∂H;
3. Second Kind: Λ(Γ) is a perfect, nowhere-dense subset of ∂H.
It turns out that
Γ is of the first kind ⇐⇒ Area(Γ\H) <∞.
All arithmetic surfaces are in this category.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
The cyclic group Γ∞ = 〈z 7→ z + 1〉 is an example of an elementary
group.
The orbits accumulate at Λ(Γ) = {∞}.
Similarly, the hyperbolic cyclic group 〈z 7→ eℓz〉 has Λ(Γ) = {0,∞}.
All other elementary groups are finite elliptic groups.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Geometric finiteness
A Fuchsian group is geometrically finite if Γ admits a fundamental
domain that is a finite-sided convex polygon.
This coincides with two other notions:
1. topological finiteness of Γ\H, and
2. Γ is finitely generated.
Spectral theory is only tractable in general for geometrically finite Γ.A theorem of Siegel says that all groups of the first kind are
geometrically finite.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Geodesics on Γ\H
If g ∈ Γ is hyperbolic, then there is a unique geodesic connecting the
two fixed points of g, called the axis of g. On its axis, g acts bytranslation by some fixed length ℓ.
In Γ\H, the axis descends to a closed geodesic of length ℓ. There is a1-1 correspondence:
closed geodesics←→ conjugacy classes of hyperbolic elements.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Cusps
The parabolic elements of Γ create cusps in Γ\H.
There is a 1-1 correspondence:
cusps←→ orbits of parabolic fixed points.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Funnels
When F meets ∂H in an interval, a funnel occurs in Γ\H.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Conical points
An elliptic fixed point of Γ causes a conical singularity in Γ\H.
There is a 1-1 correspondence:
conical points←→ orbits of elliptic fixed points.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
III. Spectral Theory
Influenced by the work of Maass, Selberg pioneered the study of the
spectral theory of hyperbolic surfaces in the 1950’s. The idea was tobring techniques from harmonic analysis into the study of
automorphic forms.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Laplacian
The Laplacian operator on H is
∆ := −y2
(∂2
∂x2+
∂2
∂y2
)
Since the Laplacian is invariant under isometries (for any metric),
∆ ◦ g = g ◦∆, for g ∈ PSL(2, R).
Hence the action of ∆ on automorphic functions is well-defined.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
The measure associated to the hyperbolic metric,
dA(z) =dx dy
y2,
is invariant under PSL(2, R) and thus defines a measure on Γ\H.
The resulting Hilbert space is L2(Γ\H, dA), with
〈 f , g〉 =
∫
F
f(z) g(z) dA(z),
for any fundamental domain F .
To define ∆ as a self-adjoint operator acting on L2(Γ\H, dA), we applythe Friedrichs extension to ∆ on the domain
D :={
f ∈ C∞0 (Γ\H) : f and ∆f ∈ L2(Γ\H, dA)
}.
(Our sign convention is ∆ ≥ 0.)
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Eigenvalues
An eigenvalue of ∆ on Γ\H is λ such that
∆φ = λφ,
where φ ∈ L2(Γ\H, dA).
In the context of automorphic forms, eigenvectors of the Laplacian arecalled Maass forms.
If Γ\H is compact, then the eigenvalues fill out the spectrum. There isan orthonormal basis of eigenfunctions {φj}, and the corresponding
eigenvalues satisfy
0 = λ0 < λ1 ≤ λ2 ≤ · · · → ∞.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Resolvent
If Γ\H is non-compact, then we understand the spectrum by studying
the resolvent (∆− λ)−1
Stone’s formula gives spectral projectors in terms of limits of as λapproaches [0,∞) from above and below.
Pα,β = limε→0
1
2πi
∫ β
α
[(∆− λ− iε)−1 − (∆− λ + iε)−1
]dλ
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
In H,
∆ys = s(1− s)ys,
suggesting that λ = s(1− s) is a natural substitution for the spectralparameter.
Indeed, if we write the resolvent as
RH(s) := (∆− s(1− s))−1,
for Re s > 12, the corresponding integral kernel is
RH(s; z, w) =1
4π
Γ(s)2
Γ(2s)cosh
−s(d/2) F(s, s; 2s; cosh
−1(d/2)),
where d := d(z, w).
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Meromorphic continuation
The same picture holds for Γ\H: the resolvent R(s) = (∆− s(1− s))−1
admits a meromorphic continuation to s ∈ C.
The set λ ∈ [0,∞) corresponds to s ∈ [ 12, 1] ∪ {Re s = 1
2}.
0 1
sλ
14
12
From the behavior of R(s) as s approaches the critical line, we can
understand the spectral projectors.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
How do we continue the resolvent? By constructing a parametrix.
1. Use R(s0) for large Re s0 in the interior.
2. In cusps and funnels, use model resolvents for cylindrical
quotients to construct R0(s).
3. Paste these together using cutoffs χj, to get the parametrix
M(s) := χ2R(s0)χ1 + (1− χ0)R0(s)(1 − χ1),
and compute the error
(∆− s(1− s))M(s) = I − K(s).
4. Show that K(s) is compact on a weighted L2 space, and use the
Analytic Fredholm Theorem to invert
R(s) = M(s)(I − K(s))−1.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Spectrum of ∆
From the structure of the resolvent, we can deduce basic spectral
information.
1. For Γ\H compact, ∆ has discrete spectrum in [0,∞).
2. For Γ\H non-compact, ∆ has absolutely continuous spectrum[ 1
4,∞) and discrete spectrum contained in [0,∞).
3. If Γ\H has infinite-area, then there are no embedded eigenvalues,
i.e. the discrete spectrum is finite and contained in (0, 14).
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Resonances
The poles of the meromorphically continued resolvent R(s) are calledresonances.
These include points with Re s ≥ 12
for which λ = s(1− s) is a discrete
or embedded eigenvalues, and possibly other points with Re s < 12.
-10
10
-10
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Cusp forms
For simplicity, assume that Γ\H has finite area with a single cusp. We
can assume that the cusp corresponds to the point∞ ∈ H, withΓ∞ := 〈z 7→ z + 1〉 ⊂ Γ and F bounded by |Re s| = 1
2.
12
− 12
The cusp forms are defined by
Hcusp :=
{f ∈ L2(F , dA) :
∫ 1
0
f dx = 0 for a.e. y > 0
}.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
In other words, if we expand f in a Fourier series
f(z) =∑
cn(y)e2πinx,
the n = 0 coefficient vanishes for y > 0.
For solutions of (∆− s(1− s))f = 0 the coefficient c0(y) behaves like ys
or y1−s as y→∞, while the non-zero modes are Bessel functions thateither grow or decay exponentially in y.
This fact implies that the restriction of ∆ to Hcusp has purely discretespectrum.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Maass cusp forms
The eigenvectors of the restriction of ∆ to Hcusp are called Maass cusp
forms.
Embedded eigenvalues (λ ≥ 14) are automatically cusp forms. For
λ < 14, eigenvalues may or may not be cusp forms.
Selberg showed that certain arithemetic surfaces have an abundance
of Maass cusp forms. But Phillips and Sarnak showed that thesedisappear when the arithmetic surface is deformed to an ordinary
hyperbolic surface. They conjectured that Hcusp is small or empty for a
general cofinite Fuchsian group.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Eisenstein series
(We continue to assume one cusp at∞ with Γ∞ ⊂ Γ.)
Since ys solves the eigenvalue equation and is invariant under Γ∞, we
can try to solve the eigenvalue equation on Γ\H by averaging
E(s; z) :=∑
g∈Γ∞\Γ
(Im gz)s
=∑
g∈Γ∞\Γ
ys
|cz + d|2s.
This is called an Eisenstein series. It converges for Re s > 1, but by
connecting it to R(s), we can show that it extends meromorphically to
s ∈ C.
Eisenstein series give a way to parametrize the continuous spectrum.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Spectral decomposition
The residual spectrum consists of eigenvalues with λ < 14
which do not
come from cusp forms. The span of such eigenvectors is denoted Hres.
On Hcusp ⊕Hres we have an complete eigenbasis {φj, λj} for ∆.
To complete our decomposition, define the continuous Hilbert spaceHcont = (Hcusp ⊕Hres)
⊥, so that
L2(Γ\H, dA) = Hcusp ⊕Hres ⊕Hcont.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
The spectrum of ∆ on Hcont is purely continuous, and Eisenstein series
give the spectral decomposition on this subspace.
This means that general f ∈ L2(Γ\H, dA) can be writen
f(z) =
∫ ∞
−∞
E( 12
+ ir; z)a(r) dr +∑
bjφj(z),
where
a(r) :=
∫
F
E( 12
+ ir; z)f(z) dA(z),
bj := 〈φj, f〉.
E(s; z) is the cusp analog of a plane wave.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Scattering matrix
As y→∞,E(s; z) = ys + ϕ(s)y1−s + O(y−∞),
for a meromorphic function ϕ(s) called the scattering matrix (or
scattering determinant, in this one-cusp case).
The poles of ϕ(s) are the scattering poles. These include points whereλ = s(1− s) is in the residual spectrum.
Cusp are resonances but not scattering poles.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
resonances = scattering poles + cusp resonances
-1.0 -0.5 0.5 1.0
-20
-10
10
20
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Multiple cusps
If there are many cusps, for each cusp we consider a model where the
corresponding parabolic fixed point is moved to∞. The y coordinatefor the j-th cusp is denoted yj.
For cusp forms, the vanishing condition applies separately to eachcusp.
For cusp i, the Eisenstein series Ei(s; z) is defined as an average of yi
over Γ. We then consider the asymptotic expansion in the j-th cusp:
Ei(s; z) ∼ δijys + ϕij(s)y
1−sj .
The scattering matrix is now an actual matrix [ϕij], with scatteringdeterminant ϕ(s).
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Infinite-area case
The presence of a single funnel means there can be no cusp forms. Inparticular, the discrete spectrum is finite and contained in (0, 1
4).
There are no embedded eigenvalues.
We can still parametrize the continuous spectrum by the analog of
Eisenstein series. These analogs are now called Poisson operators.
Asymptotic expansions of Poisson operators define the scattering
matrix, for which the funnel components are pseudodifferentialoperators.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
IV. Selberg trace formula
The central result in the spectral theory of hyperbolic surfaces is theSelberg trace formula.
To any f ∈ C∞[0,∞) we can try to define an operator Kf with integralkernel
Kf (z, w) :=∑
g∈Γ
f(d(z, gw)).
(The sum will only converge if f decays sufficiently at∞.)
The Selberg trace formula computes the trace of Kf in two different
ways.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Traces in the compact case
Suppose Γ\H is compact, with spectrum {λj} and eigenvectors {φj}.
Since Kf is a smoothing operator on Γ\H, the trace could be written as
tr Kf =
∫
Γ\H
Kf (z, z) dA(z).
On the other hand, if the eigenvalues of Kf are {κj}, then
tr Kf =∑
j
κj.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Spectral trace computation
Many geometric operators have the same form as Kf , such as the
resolvent and heat operators.
Kf always commutes with ∆ is diagonalized by the basis {φj}.
To compute κj, consider the eigenvalue equation
κjφj(w) =
∫
F
Kf (w, z) φj(z) dA(z)
=∑
g∈Γ
∫
F
f(d(w, gz)) φj(z) dA(z)
=
∫
H
f(d(w, z)) φj(z) dA(z)
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Now set w = i and sj =√
λj − 1/4.
We can exploit the fact that ysj solves the same eigenvalue equation as
φj,
(∆− λj)ysj = 0,
to prove
κjφj(i) =
∫
H
f(d(i, z)) φj(z) dA(z) = φj(i)
∫
H
f(d(w, z)) ysj dA(z).
This gives κj in terms of f ,
κj =
∫
H
f(d(i, z)) ysj dA(z)
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Length trace computation
Assume now that Γ\H is smooth and compact. This means that Γcontains only hyperbolic elements. Recall that closed geodesics of Γ\Hcorrespond to conjugacy classes of hyperbolic elements.
The length trace computation starts from
tr Kf =
∫
F
Kf (z, z) dA(z)
=∑
g∈Γ
∫
F
f(d(z, gz)) dA(z).
The trick is to organize the sum over Γ as a sum over conjugacy
classes, and then express these in terms of lengths of closed geodesics.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Decomposition of Γ
Let Π be a complete list of inconjugate primitive elements of Γ. Thenwe can write
Γ− {I} =⋃
g∈Π
⋃
k∈N
⋃
h∈Γ/〈g〉
{hgkh−1}.
Associated to each g ∈ Π is a primitive closed geodesic of Γ\H, and
the corresponding lengths form the primitive length spectrum:
L(Γ) := {ℓ(g) : g ∈ Π}.
(Note that g and g−1 are not conjugate; this is the ‘oriented’ lengthspectrum.)
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Sum over Γ
Using the conjugacy class decomposition of Γ, we write the trace as
tr Kf = f(0) Area(Γ\H) +∑
g∈Π
∑
k∈N
∑
h∈Γ/〈g〉
∫
F
f(d(z, hgkh−1z)) dA(z)
By a change of variables, we can write
∫
F
f(d(z, hgkh−1z)) dA(z) =
∫
hF
f(d(z, gkz)) dA(z)
The union of hF over Γ/〈g〉 is a fundamental domain for the cyclic
group 〈g〉. We could replace this by any other fundamental domain, so
∑
h∈Γ/〈g〉
∫
F
f(d(z, hgkh−1z)) dA(z) =
∫
F〈g〉
f(d(z, gkz)) dA(z).
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
We can conjugate g to z 7→ eℓz and take F〈g〉 = {1 ≤ y ≤ eℓ}, so that
∑
h∈Γ/〈g〉
∫
F
f(d(z, hgkh−1z)) dA(z) =
∫
{1≤y≤eℓ}
f(d(z, ekℓz)) dA(z).
The integral evaluates to
ℓ
sinh(kℓ/2)
∫ ∞
kℓ
f(cosh t)√2 cosh t− 2 cosh kℓ
sinh t dt.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
With this computation, we have
tr Kf = f(0) Area(Γ\H)
+∑
ℓ∈L(Γ)
∑
k∈N
ℓ
sinh(kℓ/2)
∫ ∞
kℓ
f(cosh t)√2 cosh t− 2 cosh kℓ
sinh t dt.
Recall that the spectral computation gave
tr Kf =
∞∑
j=0
h(√
λj − 14
),
where
h(r) :=
∫
H
f(d(i, z)) yr dA(z).
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Selberg trace formula (smooth compact case)
In terms of the function h,
∞∑
j=0
h(√
λj − 14
)=
Area(Γ\H)
4π
∫ ∞
−∞
rh(r) tanh πr dr
+∑
ℓ∈L(Γ)
∑
k∈N
ℓ
sinh(kℓ/2)h(kℓ).
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
General finite area case
The Trace Formula is somewhat trickier to prove for non-compact
Γ\H, because the operator Kf is not trace class. And the integral
∫
Γ\H
Kf (z, z) dA(z)
diverges!
To prove the trace formula, we must restrict operators to yj ≤ N in
each cusp and then carefully take the limit N →∞.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
On the spectral side, extra ‘scattering’ terms appear to account for the
continuous spectrum:
∑
j
h(√
λj − 14
)− 1
2π
∫ ∞
−∞
ϕ′
ϕ( 1
2+ ir)h(r) dr + 1
2h(0) tr[ϕij(
12)],
where [ϕij(s)] is the scattering matrix and ϕ(s) the scattering
determinant.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
On the length side, our decomposition of Γ must include conjugacy
classes of primitive elliptic and parabolic elements, as well as
hyperbolic.
For each elliptic fixed point of order m we pick up a term
m−1∑
k=1
1
m sin(πk/m)
∫ ∞
−∞
e−2πkr/m
1− e−2πrh(r) dr.
For each cusp, we add a term
−1
π
∫ ∞
−∞
Γ′
Γ(1 + ir)h(r) dr + 1
2h(0)− h(0) log 4
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Selberg zeta function
Many applications of the theory come through the Selberg zeta
function,
Z(s) :=∏
ℓ∈L(Γ)
∞∏
k=1
(1− e−(s+k)ℓ
).
Roughly speaking, the logarithmic derivative of Z(s) is what appears
on the length side when we take the trace of the resolvent.
The product converges for Re s ≥ 1, but spectral methods can be used
to prove that Z(s) has a meromorphic extension.
One nice feature of Z(s) is that its zeros are essentially the resonances.(There are some extra ‘topological’ poles and zeros.)
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Applications of the trace formula
For the finite-area case:
1. The resonance set and the length spectrum determine each other,and also χ and number of cusps.
2. Prime Geodesic Theorem:
#{
eℓ ≤ x}∼ Li x +
∑Li(xsj),
where {sj(1− sj)} are the eigenvalues in (0, 14).
3. Weyl-Selberg asymptotic formula
#{|λj| ≤ r
}− 1
4π
∫ √r−1/4
−√
r−1/4
ϕ′
ϕ( 1
2+ it) dt ∼ Area(Γ\H)
4πr
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
V. Arithmetic surfaces
The term arithmetic implies a restriction to integers.
Consider a finite dimensional representation ρ : PSL(2, R)→ GL(n, R).Restricting to integer entries gives a Fuchsian group,
Γ :={
g ∈ PSL(2, R) : ρ(g) ∈ GL(n, Z)}.
Such groups, along with their subgroups of finite index, are calledarithmetic Fuchsian groups.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Modular group
The prototype of an arithmetic Fuchsian group is the modular group
ΓZ := PSL(2, Z).
The group is generated by the elements
T =
(1 10 1
), S =
(0 1−1 0
),
or
T : z 7→ z + 1, S : z 7→ −1
z.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
The standard fundamental domain for ΓZ is
F ={
z ∈ H : |Re z| ≤ 12, |z| ≥ 1
}
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Modular surface
The modular surface is X := ΓZ\H.
The generator S fixes i, and the point p = eiπ/3 is fixed by ST−1p.
ii
pp
Gauss-Bonnet gives Area(X) = π3
.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Congruence subgroups
For N ≥ 1 the principal congruence subgroup of level N is
ΓZ(N) :={
g ∈ ΓZ : g ≡ I mod N}
For example
ΓZ(2) =
{(odd eveneven odd
)⊂ PSL(2, Z)
}
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
The quotients are the modular surfaces,
X(N) := ΓZ(N)\H.
The surface X(2) is a sphere with 3 cusps and area 2π.
The geometry gets more complicated as N increases - the genus ofX(N) is approximately N3 for N large.
Number theorists are particularly interested in Maass cusp forms onX(N).
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Spectral theory of the modular surface
Let continue with ΓZ and the modular surface X.
Recall that the scattering matrix was defined in terms of the Eisenstein
series
E(s; z) =∑
Γ∞\ΓZ
ys
|cz + d|2s.
In this case, we can compute the asymptotic expansion explicitly,giving the scattering matrix
ϕ(s) =√
πΓ(s− 1
2)
Γ(s)
ζ(2s− 1)
ζ(2s),
where ζ(z) is the Riemann zeta function. Meromorphic continuation isthus clear, and the scattering poles are solutions of ζ(2s) = 0.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
The modular surface also has many cusp forms:
-1.0 -0.5 0.5 1.0
-20
-10
10
20
Cusp eigenvalue λj = 14
+ r2j −→ resonances at s = 1
2± irj.
Riemann zeros −→ scattering poles on Re s = 14.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
The Selberg trace formula implies that
#{
cusp resonances with |s− 12| ≤ t
}∼ t2
6.
In contrast, the asymptotics of the Riemann zeros gives
#{
scattering poles with |s− 12| ≤ t
}∼ 2t log t
π.
The cusp forms dominate the spectrum.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Hecke operators
Why makes arithmetic surfaces so special?
They have hidden symmetries, called Hecke operators.
For f ∈ L2(ΓZ\H, dA) and n ∈ N, define
Tn f(z) :=1√n
∑
ad=n
d−1∑
b=0
f
(az + b
d
).
It’s clear that Tn commutes with ∆. What is not so obvious is that Tn f
is still invariant under ΓZ.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
As an example, consider
T3 f(z) =1√3
[f(3z) + f
( z
3
)+ f
(z + 1
3
)+ f
(z + 2
3
)].
Invariance under z 7→ z + 1 is actually obvious.
The other generator is z 7→ −1/z, so consider
T3 f
(−1
z
)=
1√3
[f
(−3
z
)+ f
(− 1
3z
)+ f
(z− 1
3z
)+ f
(2z− 1
3z
)].
The first 2 terms just switched places:
f
(−3
z
)= f
( z
3
), f
(− 1
3z
)= f(3z).
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
For the other terms we can use the matrix identity,
(1 −1
3 0
)=
(1 −1
3 −2
) (1 2
0 3
)
(note that det
(1 −13 −2
)= 1!), to see that
f
(z− 1
3z
)= f
(z + 2
3
).
Similarly,
f
(2z− 1
3z
)= f
(z + 1
3
).
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Why does this work?
Let
Υn := {A ∈ GL(2, Z) : det A = n},and define an equivalence relation on Υn,
A1 ∼ A2 if A1 ∈ ΓZA2.
List the matrices occurring in the definition of Tn:
An =
{(a b
0 d
): ad = n, 0 ≤ b ≤ d− 1
}.
Claim: The partition of Υn into distinct equivalence classes is
Υn =⋃
A∈An
[ A]
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Consider
Tn f(z) =1√n
∑
A∈An
f(Az).
Suppose g ∈ ΓZ and A ∈ An. Then det(Ag) = n, so Ag ∈ Υn. Our claim
then says that
Ag = hA′,
for h ∈ ΓZ and A′ ∈ An.
If f is automorphic,
f(Agz) = f(hA′z) = f(A′z),
so
Tn f(gz) =1√n
∑
A′∈An
f(A′z) = Tnf(z).
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
The full ring of Hecke operators Tn together with ∆ can besimultaneously diagonalized.
In particular, we can choose a basis of Maass cusp forms such that
∆φj = λjφj, Tnφj = τj(n)φj.
Indeed, it is conjectured that the cusp spectrum of X is simple, which
would imply that Maass cusp forms are automatically Hecke
eigenfunctions.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
What do the Hecke symmetries look like? Here’s a picture for T3.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Compare to density plots of Maass cusp forms:
(Alex Barnett/Holger Then)
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
L-function connection
One primary reason that Maass cusp forms are important in number
theory is the connection to L-functions (series which generalize theRiemann zeta function).
If φ is a Maass cusp form, with Hecke eigenvalues τ(n), then we can
form a related L-series
L(s, φ) :=∑
n
τ(n)n−s =∏
p
1
1− τ(p)p−s − p−2s.
The L function converges for Re s large, and the modularity of φ is
equivalent to the analytic continuation of L(s, φ), with a functionalrelation connecting the values at s and 1− s.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Many spectral questions about Maass cusp forms are related to
properties of L-functions.
Artin’s conjecture connects two-dimensional representations of Galoisgroups, via the associated L-functions, to Maass cusp forms with
eigenvalue 14.
On a related note, the Selberg-Ramanujan conjecture says that X(N)has no discrete spectrum below 1
4.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Quantum ergodicity
On the high-frequency side one of the main questions has been theasymptotic distribution of cusp forms.
The geodesic flow of a hyperbolic surface is ergodic, meaning that the
only functions invariant under the geodesic flow are constant a.e.
Classical ergodicity is expected to correspond with ‘quantum
ergodicity’, meaning equidistribution of eigenvectors as j→∞.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
Quantum ergodicity is interpreted in terms of probability measuresassociated to eigenfunctions:
φj 7→ νφj= |φj|2 dA.
(There is a related phase space version, µφjon the unit tangent
bundle, called the microlocal lift).
The conjecture is that νφjapproaches dA in some sense (or µφj
approaches Liouville measure on the unit tangent bundle).
For compact manifolds with ergodic geodesic flow, this is known to be
true for ‘most’ subsequences jk →∞. Quantum unique ergodicity is
the question of whether it holds for all sequences.
For arithmetic surfaces, QUE becomes a problem of estimating certainspecial values of L-functions, and it is this connection that has lead to
a proof in this case.
Hyperbolic geometry Fuchsian groups Spectral theory Selberg trace formula Arithmetic surfaces
General references:
1. S. Katok, Fuchsian Groups, University of Chicago Press, Chicago,IL, 1992.
2. P. Buser, Geometry and Spectra of Compact Riemann Surfaces,
Birkhauser Boston Inc., Boston, MA, 1992.
3. A.B. Venkov, Spectral Theory of Automorphic Functions and Its
Applications, Kluwer Academic Publishers, Dordrecht, 1990.
4. D. Borthwick, Spectral Theory of Infinite-Area Hyperbolic Surfaces,Birkhauser Boston Inc., Boston, MA, 2007.
5. T.M. Apostol, Modular functions and Dirichlet series in number
theory. Springer-Verlag, NY, 1990
6. P. Sarnak, Spectra of hyperbolic surfaces, lecture notes, 2003.
7. P. Sarnak, Recent progress on QUE, lecture notes, 2009.